### Nuprl Lemma : exp-exists-ext

`∀x:ℝ. ∃a:ℝ. Σn.(x^n)/(n)! = a`

Proof

Definitions occuring in Statement :  series-sum: `Σn.x[n] = a` rnexp: `x^k1` int-rdiv: `(a)/k1` real: `ℝ` fact: `(n)!` all: `∀x:A. B[x]` exists: `∃x:A. B[x]`
Definitions unfolded in proof :  canonical-bound-property rmul_preserves_rleq r-archimedean ratio-test-ext rleq_functionality r-archimedean2 iff_weakening_equal exp-series-converges exp-exists guard: `{T}` implies: `P `` Q` all: `∀x:A. B[x]` sq_type: `SQType(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` reg-seq-mul: `reg-seq-mul(x;y)` bnot: `¬bb` le_int: `i ≤z j` reg-seq-adjust: `reg-seq-adjust(n;x)` reg-seq-inv: `reg-seq-inv(x)` canonical-bound: `canonical-bound(r)` imax: `imax(a;b)` accelerate: `accelerate(k;f)` eq_int: `(i =z j)` btrue: `tt` bfalse: `ff` mu-ge: `mu-ge(f;n)` rinv: `rinv(x)` int-to-real: `r(n)` rmul: `a * b` rdiv: `(x/y)` absval: `|i|` lt_int: `i <z j` ifthenelse: `if b then t else f fi ` quick-find: `quick-find(p;n)` rlessw: `rlessw(x;y)` rminus: `-(x)` so_lambda: `λ2x.t[x]` member: `t ∈ T`
Lemmas referenced :  int_subtype_base subtype_base_sq exp-exists canonical-bound-property rmul_preserves_rleq r-archimedean ratio-test-ext rleq_functionality r-archimedean2 iff_weakening_equal exp-series-converges
Rules used in proof :  independent_functionElimination dependent_functionElimination natural_numberEquality independent_isectElimination intEquality cumulativity isectElimination equalitySymmetry equalityTransitivity sqequalHypSubstitution thin sqequalRule hypothesis extract_by_obid instantiate cut sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution introduction

Latex:
\mforall{}x:\mBbbR{}.  \mexists{}a:\mBbbR{}.  \mSigma{}n.(x\^{}n)/(n)!  =  a

Date html generated: 2018_05_22-PM-02_04_08
Last ObjectModification: 2018_05_21-AM-00_16_53

Theory : reals

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